I came across this interesting article by H. Allen Orr comparing the mathematics of fitness and of portfolio theory (to access the article you will need to be on campus or hooked in the Internet via a VPN Client. Otherwise I put a pdf of the article in our Week 1 reading folder). I was initially intrigued because it presents a link between economics and natural selection. And, as it turns out, the content of the article is on something I’ve written about in the past.

The key concept here is variance. Consider two traits whose long term expected outcome differs only by their variance. Under certain conditions the trait with a smaller variance will have a greater long term relative frequency. J. Gillespie has a series of seminal articles about this in the 1970s. H. Allen Orr tells us that the same results have been discussed in the investment literature. Orr’s presentation is particularly lucid. It is worth the price of admission.

Zac Ernst and I have recently published a paper on the “Gillespie effect”. You can find it here–if you can access the journal–or if not, you can find it here.

Zac and I think that the Gilespie effect is significant because it demonstrates that, unlike the prevailing view among philosophers of biology– fitness is not sufficiently defined as a dispositional property. In the Gillespie case the main determinate for fitness is a purely statistical effect, not a dispositional property of the organisms. Orr doesn’t say anything about this.

Zac and my paper are an outgrowth of a paper I wrote with Richard Lewontin a few years ago. Here’s that paper.

I’m interested in your reaction to all this.

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It is nearly 1 AM and I have read Orr three times. I haven’t done the necessary step of writing out the formal argument. I’m too tired tonight and I think I need to tie that formalization to a like analysis of Ariew and Ernst and to the Price Equation. But here is what bugs me now. Orr’s analysis of he portfolio problem is as basic as it gets — no improvement on Bernoulli. Most individuals maximize their utility over a series of risky choices by finding their own idiosyncratic point of optimality between expected return (income, wealth) and the risk (dispersion around the expected outcome) A PRIORI. Ariew is moderately risk averse: drives a Volvo, buys blue chip stocks and places savings in certificates of deposit. Westgren is less risk averse: drives a Viper, invests in tech stocks and places savings in lottery tickets. A third person, Kaczynski, buries his wealth in paint cans in the back yard. It’s pretty clear that Kaczynski is much more risk averse than Ariew and Westgren and is willing to take ZERO expected return to have ZERO risk. Buried money earns no interest. And it’s pretty clear that Westgren accepts more absolute risk than Ariew. But there is nothing to say that the expected return of Westgren’s portfolio is greater than Ariew’s. Should be, if their risk/return tradeoffs are the same AND if they have identical choices from which to draw their portfolios, but these constraints may not be valid.

Now, if I cross the metaphorical divide after Orr, I wonder if holding the fitness of A2 constant over the two time periods while examining the change in absolute fitness in A1 creates a peculiarly shaped risk/return tradeoff when the allele frequencies are not independent?

And I need help in understanding something. The X axis is a line with infinite points, each representing a unique combination (portfolio) of the risky assets, with each point having (effectively) a unique expected return. They are somehow each measured and then placed in order of increasing expected return. The curves in the figure represent the utility of a particular individual to the return associated with each point on the X axis. The risk-averse curve shows that there is lower utility for higher payoffs because there is IMPLIED higher risk associated with higher returns. What is the analogous model in allele fitness? What is on the X axis — relative A1/A2 allele frequencies from 0/100 to 100/0? If so, is the Y axis total population fitness? If so, what do we need to know and what do we need to believe to draw a (smooth?) curve that relates the X axis points to the Y axis?

More later, but I need your help with this.

Randy: I gather what you are trying to do is to work out the analogy between the portfolio case and fitness. One of your concerns is that the effect is purely mathematical and whether it applies depends on conditions. Agreed.

And, this brings up an interesting conceptual distinction. First, there’s “pure math” as in “unrealistic expectation”, in other words, it won’t ever happen in real life. Second, there’s “pure math” in that the explanation for a real world effect is purely mathematical. It is the latter that interests me.

As for the former, I think it is pretty easy to construct an applicable case. This comes from my paper with Lewontin. Imagine two types whose propensities are identical except that they have different reproductive schedules. The first produces two babies every generation. The second produces either one or three babies (at random) every generation. Start with two babies (initial condition). The expected relative frequency of the two types are:

A: (.5)(4/6 + 4/10) = .535

B: (.5)(2/6 + 2/10) = .465

That’s an application of the Gilespie effect. The one with the skinnier variance has a higher expected relative frequency.

It is pretty easy to intuit why this is the case: “the slow and steady win the race”.

I don’t know exactly how to answer your question about the x-axis and y-axis for the fitness case. But, I don’t think it would be too hard to replot the points in terms of expected relative frequency of types vs. generational time (like I did above).

Thanks, André. I’ll go back to your paper with Lewontin.

The next step in working through the analogy is that the no variance case (2 babies per year) is called the Certainty Equivalent in portfolio theory. The risky reproductive schedule is the Lottery. The shape of the utility function determines how much of a premium (earnings) that the portfolio holder needs to earn to choose the lottery. (It’s a simple graph. I’ll try to replicate it in a PPT and post it.) Again, it is all about expected payoffs — how much does the average quarterly expected earnings from shares of a stocks have to be before I take money out of my savings account and buy the shares?

Now, if the evolutionary analogy is any more than a purely mathematical explanation, there should be a meaning for the “reproductive risk premium” that makes your A and B equally likely to be “chosen”. Some may call it Fitness.

The step beyond is adding alleles/adding investment alternatives (gold, Old Masters, government bonds, baseball cards,…). In the latter case, we get an n-dimensional depiction of all possible portfolios and the utility function in n-space “chooses” the optimal. Of course, as the dimensionality increases, we are likely to have multiple optima. In the former case, we get genotype space and the natural selection process chooses an optimum (or multiple optima).

Because I have a hard time perceiving n-space, I can project n-dimensions on to three dimensions or two dimensions to obtain a map (without explicit dimensions) of the portfolio choices and their expected payoffs adjusted for risk. Or I get Sewall Wright’s fitness landscape.

Maybe.